Integral models of certain Shimura curves

Kevin Buzzard UC Berkeley, Berkeley, CA.

Original version 13/5/1996. Some typos fixed 14/6/2002.

Integral models of certain Shimura curves

Kevin Buzzard†

§0. Introduction.

Much of the theory of the mod p reduction of modular curves is now well-understood and well-documented (for example in [KM]). Although a lot is also known to the experts in the more general setting of curves attached to arbitrary indefinite quaternion algebras over ℚ, less seems to have been written down. In particular, the analogues of the results of Deligne and Rapoport on the bad reduction of X_{0}(p) and X_{1}(p) seem not to be in the literature. Over totally real fields other than ℚ, one has the article [C] of Carayol, but the case of ℚ is excluded in [C]. Theorem 4.7 and Theorem 4.10 of this paper are the analogues of the results of Deligne and Rapoport in this more general setting, and the results show a strong analogy with the modular curve case, as expected.

§1. False elliptic curves.

Fix an indefinite non-split quaternion algebra D over ℚ, and let d = disc(D). By definition, d is the product of the (finitely many) primes p for which D ⊗_{ℚ} ℚ_{p} is not isomorphic to M_{2}(ℚ_{p}). Fix once and for all a maximal order O_{D} of D.

Let R = \underleftarrow{lim} (ℤ/Mℤ), where the limit is over all positive integers M prime to d. Fix once and for all an isomorphism κ : O_{D} ⊗_{ℤ} R → M_{2}(R), and note that κ induces an isomorphism O_{D} ⊗_{ℤ} T ≅ M_{2}(T) for all quotients T of R. Write O_{D,f} for O_{D} ⊗_{ℤ} ℤ̂. Then κ induces a natural map O_{D,f} → M_{2}(ℤ/Nℤ) for any positive integer N prime to d, by the composite of the maps O_{D,f} → O_{D} ⊗_{ℤ} R →^{κ} M_{2}(R) → → M_{2}(ℤ/Nℤ). Finally, fix once and for all an isomorphism κ_{∞} : D ⊗ ℝ → M_{2}(ℝ).

Let S be a scheme on which d is invertible. A false elliptic curve over S is a pair (A/S, i) where A/S is an abelian surface (that is, A is an abelian scheme over S of relative dimension two), and i : O_{D} ↪ End_{S}(A) is an injective ring homomorphism (where throughout this paper all ring homomorphisms are assumed to be identity-preserving). We will frequently abuse notation and talk of “the false elliptic curve A/S ”, or even just “the false elliptic curve A”. Strictly speaking we should really refer to “false elliptic curves for D”, but because we shall never vary our choice of D it will hopefully cause no confusion if we omit it. Also, we shall only ever refer to false elliptic curves over ℤ[1/d]-schemes. Some references for false elliptic curves are [B], [BC], Chapter 4 of [DT] and Chapter 1 of [R]. If A/S is a false elliptic curve, and T is an S-scheme, the pullback A_{T} = A ×_{S} T can be naturally given the structure of a false elliptic curve over T.

† An initial version of this paper was part of the Cambridge PhD thesis of the author. This version was written while the author was at the IAS in Princeton, and he would like to thank Conrad, Deligne, Edixhoven, Katz and the referee for several helpful remarks. The author was supported by a Trinity College Junior Research Fellowship in Cambridge, and partially supported by NSF grant DMS 9304580 whilst at the IAS.

Let x ↦ x^{ι} denote the canonical involution of D. Fix t ∈ O_{D} with t^{2} = −d. Such t exist because ℚ(\sqrt{−d}) splits D. Define another involution ∗ on D by x^{∗} = t^{−1}x^{ι}t. A local calculation shows that O_{D} is stable under ∗. We now observe that if (A/S, i) is a false elliptic curve, then there is a unique principal polarisation on A such that if \overline{s} is a geometric point of S, then the corresponding Rosati involution on End(A\overline{s}) induces ∗ on O_{D}. This is proved for S the spectrum of an algebraically closed field of characteristic 0 in [BC] III.1.5. Moreover, the proof extends to characteristic p not dividing d, after one observes that if A/k is a false elliptic curve over an algebraically closed field of characteristic p, then the p-divisible group A[p^{∞}] is of the form G × G, with O_{D} acting via the natural action of M_{2}(ℤ_{p}). Finally, we deduce the result for a general base S/Spec(ℤ[1/d]) by the Proposition in Section 11 of [B]. We conclude that once we have fixed t, any false elliptic curve admits a canonical principal polarisation.

If (A/S, i) and (B/S, j) are false elliptic curves, then by an isogeny π : A → B of false elliptic curves we mean an isogeny π : A → B over S in the usual sense, such that j(x) ∘ π = π ∘ i(x) for all x ∈ O_{D}. If π : A → B is an isogeny of false elliptic curves, then π induces a dual isogeny π^{∨} : B^{∨} → A^{∨} which, because we have defined principal polarisations on A and B, induces a map π^{t} : B → A. One may check that this map is also an isogeny of false elliptic curves, and we refer to it as the dual isogeny to π. The composite π^{t} ∘ π : A → A is locally multiplication by an integer d, and if this integer is constant on S we call it the false degree of π. There is a canonical perfect pairing

⟨, ⟩_{π} : ker(π) × ker(π^{t}) → 𝔾_{m}.

In fact, there are two canonical perfect pairings, one being the inverse of the other. To fix our ideas, we shall use the opposite of the definition in [O]. Lemma 1.1 of [O] then shows us that if δ ∈ D, P ∈ (ker π)(S) and Q ∈ (ker π^{t})(S), then ⟨i(δ)P, Q⟩_{π} = ⟨P, j(δ^{∗})Q⟩_{π} in 𝔾_{m}(S).

We will now recall how to attach various (naïve) level structures to false elliptic curves, analogous to the level structures that one can define on elliptic curves. Let M be a positive integer prime to d, and write (O_{D} ⊗ ℤ/Mℤ)_{S} for the constant group scheme over S associated to the finite group O_{D} ⊗ ℤ/Mℤ. Note that left multiplication induces a left action of O_{D} on (O_{D} ⊗ ℤ/Mℤ)_{S}.

Definition 1.1. A naïve full level M structure on a false elliptic curve (A/S, i) is an isomorphism

α : (O_{D} ⊗ ℤ/Mℤ)_{S}⟶^{≅}A[M]

of schemes which preserves the left action of O_{D}.

Note that the existence of a full level M structure on A/S implies that M is invertible on S.

If δ ∈ O_{D}⊗ℤ/Mℤ then right multiplication by δ induces a morphism of group schemes

r_{δ} : (O_{D} ⊗ ℤ/Mℤ)_{S} → (O_{D} ⊗ ℤ/Mℤ)_{S}

which preserves the left action of O_{D}. Moreover, if δ ∈ (O_{D} ⊗ ℤ/Mℤ)^{×} then r_{δ} is an isomorphism. Hence if α is a full level M structure on a false elliptic curve A, and δ ∈ (O_{D} ⊗ ℤ/Mℤ)^{×}, then α ∘ r_{δ} is also a full level M structure on A. The map α ↦ α ∘ r_{δ} defines a left action of (O_{D} ⊗ ℤ/Mℤ)^{×} on the set of full level M structures on A.

Let H ⊆ (O_{D} ⊗ ℤ/Mℤ)^{×} be a subgroup, let S be a ℤ[1/Md]-scheme, and let (A/S, i) be a false elliptic curve. We will now define the notion of a naïve level H structure on A. Consider the contravariant functor F_{A/S,M} from S-schemes to sets, sending T/S to the set of full level M structures on A_{T} . Then F_{A/S,M} is represented by a closed subscheme of A[M], and this representing scheme is an étale (O_{D} ⊗ ℤ/Mℤ)^{×}-torsor. For each T, there is a left action of (O_{D} ⊗ ℤ/Mℤ)^{×}, and hence of H, on F_{A/S,M}(T) as defined above. If H\F_{A/S,M}(T) denotes the orbit space, then define F_{A/S,H} to be the sheafification with respect to the étale topology of the functor T ↦ H\F_{A/S,M}(T) from S-schemes to sets. Then F_{A/S,H} is represented by a quotient of the S-scheme representing F_{A/S,M} and is again a finite étale covering of S.

Definition 1.2. A naïve level H structure on (A/S, i) is an element of F_{A/S,H}(S).

The natural map O_{D,f} → O_{D} ⊗ ℤ/Mℤ induces a homomorphism O_{D,f}^{×} → (O_{D} ⊗ ℤ/Mℤ)^{×}. Let U ⊆ O_{D,f}^{×} be the preimage of H under this homomorphism. It is not difficult to check that the definition of a level H structure depends only on U, in the following sense. Let M_{1} and M_{2} be two positive integers prime to d, with M_{1} | M_{2}. Let H_{1} ⊆ (O_{D} ⊗ℤ/M_{1}ℤ)^{×} be a subgroup, and let H_{2} be its preimage in (O_{D} ⊗ℤ/M_{2}ℤ)^{×} under the natural surjection. Then the notion of an H_{1} structure and an H_{2} structure coincide for false elliptic curves over ℤ[1/M_{2}d]-schemes. Hence it makes sense to talk about the functor F_{A/S,U} and also of naïve level U structures on a false elliptic curve A/S, for U a compact open subgroup of O_{D,f}^{×}.

Finally we record how naïve level H structures may be pushed forward in some cases. Say (A/S, i) and (B/S, j) are false elliptic curves, and ψ : A → B is an isogeny of false elliptic curves. Say H ⊆ GL_{2}(ℤ/Mℤ) is a subgroup, and ϕ is a naïve level H structure on A/S. Assume moreover that the false degree of ψ is coprime to M. Then ψ induces an isomorphism A[M] → B[M] and hence composition with ψ induces an isomorphism of functors F_{A/S,M} → F_{B/S,M}. Furthermore, this isomorphism preserves the action of O_{D} and hence induces an isomorphism F_{A/S,H} ≅ F_{B/S,H}, and hence a bijection ψ_{∗} from the naïve level H structures on A/S and the naïve level H structures on B/S. In particular, ψ_{∗}ϕ is a naïve level H structure on B/S.

§2. Shimura curves.

Let M be a positive integer prime to d. Let u_{M} be the map u_{M} : O_{D,f}^{×} → GL_{2}(ℤ/Mℤ) induced by the map κ of §1.

We now make the following three definitions.

Let V_{0}(M) denote the preimage of \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ/Mℤ)|c = 0 \} under u_{M}.

Let V_{1}(M) denote the preimage of \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ/Mℤ)|c = 0 and d = 1 \} under u_{M}.

Let V_{M} be the kernel of the map u_{M}.

Let U be a compact open subgroup of O_{D,f}^{×} with the following three properties:

(i) det(U) = ℤ̂^{×} (ii) U is maximal at primes dividing d, that is, U = ∏_{p|d}(O_{D} ⊗_{ℤ} ℤ_{p})^{×} × U^{d}, where U^{d} is the projection of U onto (O_{D} ⊗_{ℤ} R)^{×} ≅ ∏_{p∤d}(O_{D} ⊗_{ℤ} ℤ_{p})^{×}. (iii) U ⊆ V_{1}(N) for some N ≥ 4 prime to d.

Not all of these assumptions will be necessary at all times, but for simplicity we shall assume them from the outset. Briefly, here are the reasons that these assumptions are here. (i) is convenient because it implies that the complex curve attached to U (see below) is connected. (ii) is much more important, as it ensures that we never consider level structures at primes dividing d, and (iii) is a smallness criterion which we shall need for later representability results. If we relaxed the condition that our objects of study be schemes, and instead allowed them to be algebraic stacks, then condition (iii) would be unnecessary. The author has applications to modular forms in mind, however, and the restrction to schemes will suffice for these purposes.

Define U_{∞} ⊆ (D ⊗ ℝ)^{×} to be the stabiliser of i = \sqrt{−1} under the natural action of (D ⊗ ℝ)^{×} ≅ GL_{2}(ℝ) on H^{±}, the union of the upper and lower half planes. Then there is a complex curve X^{D}(U)_{ℂ} whose ℂ-valued points are naturally in bijection with the set

D^{×}\(D ⊗_{ℚ} 𝔸)^{×}/U.U_{∞},

where 𝔸 denotes the ring of adèles over ℚ. Our aim is to study certain integral models of these curves. To do this is will be convenient to introduce the language of algebraic stacks, because this is the language of many of the theorems in the literature, although ultimately we shall prove that all the stacks we consider are in fact schemes. One reference for algebraic stacks is [CF], and others are mentioned therein.

Let U be a compact open subgroup of O_{D,f}^{×}, satisfying (i)–(iii) above. Let M_{U} denote the smallest positive integer M prime to d such that V_{M} ⊆ U. Let X^{D}(U) be the following category. An object of X^{D}(U) is a false elliptic curve (A/S/ℤ[1/M_{U}d], i_{A}) equipped with a level U structure α ∈ F_{A/S,U}(S), and a morphism from ((A/S, i_{A}), α) to ((B/T, i_{B}), β) in this category is a commutative diagram

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ S & ⟶ & T \end{matrix}

such that • The induced morphism A → B_{S} = B ×_{T} S is an isomorphism,

• The induced map a : End_{T} (B) → End_{S}(A) satisfies i_{A} = a ∘ i_{B}.

• α is the image of β under the natural map F_{B/T,U}(T) → F_{B_{S}/S,U}(S) = F_{A/S,U}(S). There is a morphism from X^{D}(U) to the category of ℤ[1/M_{U}d]-schemes, sending

((A/S, i_{A}), α) to S.

Theorem 2.1. X^{D}(U) is an algebraic stack over ℤ[1/M_{U}d]. Moreover, the morphism X^{D}(U) → Spec(ℤ[1/M_{U}d]) is proper and smooth of relative dimension 1.

Proof. This is proved in [B], section 14. The representability result is deduced from a theorem of M. Artin.

□

We will now use the smallness criterion (iii) on U to show that X^{D}(U) is in fact the algebraic stack associated to a scheme. For this we shall need

Lemma 2.2. Let N ≥ 4 be an integer prime to d, and let k be an algebraically closed field of either characteristic 0 or of characteristic p prime to Nd. Let (A/k, i) be a false elliptic curve over k, and let α be a V_{1}(N)-structure on A/k. Then the only automorphism of (A/k, i) which fixes α is the identity.

Proof. Because k is an algebraically closed field, the V_{1}(N)-structure α can be thought

of as an orbit of full level N structures under the action of the group H = \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) ∈

GL_{2}(ℤ/Nℤ)}. Let β : (O_{D} ⊗ ℤ/Nℤ)_{k} → A[N] be an element of this orbit. We must show that if θ is an automorphism of (A/k, i) such that θβ = βr_{δ} for some δ ∈ H, then θ is the identity. So let θ be such an automorphism.

Let ψ = θ − 1. Then ψ is an endomorphism of (A/k, i). Let G = ker(ψ). Now G is a group scheme of finite type over k, with an action of O_{D}. This action induces an action of O_{D} on G^{0}, the identity component of G, and hence on H = (G^{0})^{red}, the reduced scheme associated to G^{0}. We see that G/H is a finite group scheme over k, and that H is an abelian variety. Moreover, we have an action of O_{D} on H.

If H were 1-dimensional, then it would be an elliptic curve and hence there would be a ring homomorphism (sending 1 to 1) from O_{D} to End_{k}(H). Tensoring up by ℚ, we find a contradiction, because are no maps (sending 1 to 1) from D to ℚ, to an imaginary quadratic extension of ℚ, or to a definite quaternion algebra over ℚ.

Hence H is either zero or 2-dimensional. If H is 2-dimensional, then H = G and ψ = 0, which implies θ = 1 and we are finished. So assume for a contradiction that H = 0. Then G is finite and ψ is an isogeny. If ψ^{t} is the transpose of this isogeny, then the composite ψ^{t} ∘ ψ is multiplication by a positive integer n, the false degree of ψ. Moreover, θ is an automorphism and hence θ^{t} ∘ θ is the identity. We have

n = ψ^{t}ψ = (θ − 1)^{t}(θ − 1)
= θ^{t}θ − θ^{t} − θ + 1
= 2 − (θ^{t} + θ).

Hence (θ^{t} + θ) is multiplication by an integer, namely m = 2 − n. We deduce that θ satisfies the equation X^{2} − mX + 1 = 0 in End_{k}(A).

Now θ is an automorphism of the abelian variety A/k, commuting with the action of O_{D}. One can check that this implies that it preserves the principal polarisation defined in §1. By [M] proposition 17.5, this implies that θ has finite order in End_{k}(A). Say θ has order f. Then set C = ℤ[θ] ⊆ End_{k}(A), the ring generated by θ in End_{k}(A). Now C ⊗ ℚ must be a quotient of ℚ[T]/(T^{f} − 1), and the fact that θ^{2} − mθ + 1 = 0 and that C ≠ 0 implies that T^{2} − mT + 1 and T^{f} − 1 must have a common factor in ℚ[T]. Hence one of the roots of T^{2} − mT + 1 = 0 is a root of unity and hence |m| ≤ 2. But by definition, m = 2 − n and hence 1 ≤ n ≤ 4 because n is a positive integer.

The k-valued point P = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) of (O_{D} ⊗ ℤ/Nℤ)_{k} is mapped by β to a k-valued point Q of A. We observe that β = θ^{t}θβ = θ^{t}βr_{δ} and hence that θ^{t}β = βr_{δ^{−1}}. Hence mβ = θβ + θ^{t}β = β(r_{δ} + r_{δ^{−1}}) = β(r_{δ+δ^{−1}}). Applying these maps to P, we deduce that mQ = 2Q and hence nQ = 0 ∈ A[N](k). Because β is an isomorphism we deduce that nP = 0 in M_{2}(ℤ/Nℤ) and hence that N divides n. If N ≥ 5 then this is the desired contradiction, so we can assume henceforth that N = 4. Because N divides n, we must have n = 4 and m = −2, so θ^{2} + 2θ + 1 = 0. Hence θ + 1 is certainly not an isogeny, as its square is zero. We deduce that θ + 1 must be zero, and so θ = −1. So β = −βr_{δ}, and applying these maps to P we deduce that Q = −Q and hence that Q is a point of order 2, a contradiction because P is of order 4.

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Corollary 2.3. If U ⊆ O_{D,f}^{×} is open and compact, and satisfies properties (i)– (iii) above, then X^{D}(U) is the stack associated to a regular scheme, and the morphism X^{D}(U) → Spec(ℤ[1/M_{U}d]) is projective.

Proof. Firstly we show that X^{D}(U) is the stack associated to an algebraic space. To do this, it suffices by Example 4.9 of [DM] to show that the objects of the category X^{D}(U) have no non-trivial automorphisms. Assume for a contradiction that A/S is an abelian scheme with a level U structure α, and there is an automorphism θ ≠ 1 preserving α. By [DR] II 1.14 there is a geometric point k → S of S such that θ_{k} : A_{k} → A_{k} is not the identity. But this contradicts Lemma 2.2, because U ⊆ V_{1}(N) for some N ≥ 4. Hence X^{D}(U) is an algebraic space.

Now ℤ[1/M_{U}d] is regular, and the morphism X^{D}(U) → Spec(ℤ[1/M_{U}d]) is smooth of relative dimension 1, and proper. It follows that X^{D}(U) → Spec(ℤ[1/M_{U}d]) is quasiprojective, and hence by [Kn] (II.7.6) we see that X^{D}(U) is a scheme. Finally, the map X^{D}(U) → Spec(ℤ[1/M_{U}d]) is quasiprojective and proper, hence projective by [Kn] II.7.8.

□

Finally, we record two useful propositions.

Proposition 2.4. If U satisfies (i)–(iii) above, then the fibres of the morphism X^{D}(U) → Spec(ℤ[1/M_{U}d]) are geometrically irreducible.

Proof. By the theory over ℂ (see [Gi]), the generic fibre of X^{D}(U) is geometrically connected. By Zariski’s connectedness theorem, all the fibres of X^{D}(U) → Spec(ℤ[1/M_{U}d]) are geometrically connected. By [B] §16, Corollary, they are geometrically unibranch and hence geometrically irreducible.

□

Proposition 2.5. If U_{1} ⊆ U_{2} both satisfy (i)–(iii) above, then the canonical forgetful morphism X^{D}(U_{1}) → X^{D}(U_{2}) over Spec(ℤ[1/M_{U_{1}}d]) is étale.

Proof. Clear.

§3. New structures at l.

We now consider more carefully the l-power torsion in false elliptic curves, where throughout this section l is a prime not dividing d. Let A/S be a false elliptic curve over a ℤ[1/d]-scheme S. Then the l-divisible group A[l^{∞}] attached to A inherits an action of O_{D} and hence of O_{D} ⊗_{ℤ} ℤ_{l}, which via the map κ of §1 can be identified with M_{2}(ℤ_{l}).

If we set ẽ = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(ℤ_{l}) then A[l^{∞}] splits as ker(ẽ) × ker(1 − ẽ). Moreover,

ker(ẽ) and ker(1 − ẽ) are isomorphic, because ẽ and 1 − ẽ are conjugate in M_{2}(ℤ_{l}). More

specifically, the automorphism of A[l^{∞}] induced by the element ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ∈ M_{2}(ℤ_{l}) induces

an isomorphism ker(ẽ) → ker(1 − ẽ). Henceforth we shall identify ker(1 − ẽ) with ker(ẽ) via this isomorphism.

If S = Spec(k) is the spectrum of an algebraically closed field, then we can completely classify the possibilities for ker (ẽ), as follows.

• If the characteristic of k is not equal to l, then A[l^{∞}] ≅ (ℚ_{l}/ℤ_{l})^{4} and hence ker(ẽ) ≅ (ℚ_{l}/ℤ_{l})^{2}.

• If the characteristic of k is equal to l, then there are two possibilities. A is an abelian variety and it is a classical result that the l-divisible group A[l^{∞}] is isomorphic to (μ_{l^{∞}})^{r} × (ℚ_{l}/ℤ_{l})^{r} × Γ, where Γ/k is a local-local l-divisible group. Because we have seen that A[l^{∞}] ≅ G × G for some l-divisible group G, we deduce that either r = 2 and G = μ_{l^{∞}} × ℚ_{l}/ℤ_{l}, or r = 0 and G is the unique one-parameter formal Lie group over k of height 2. In particular, in the former case we have A[l^{∞}] ≅ (E[l^{∞}]^{2}) for E/k any ordinary elliptic curve, and in the latter case A[l^{∞}] ≅ (E[l^{∞}]^{2}) for E/k any supersingular elliptic curve. Note that in either case there is an elliptic curve E/k with A[l^{∞}] ≅ (E[l^{∞}]^{2}), a fact which will be very useful to us later, because it will enable us to deduce results about the local rings of the Shimura curves we are studying from facts about the local rings of modular curves.

If A/k is a false elliptic curve over any field of characteristic l, then we have just seen above that the l-divisible group of A\overline{k} is isomorphic to E[l^{∞}]^{2} for E/\overline{k} some elliptic curve. We shall call A/k ordinary or supersingular depending on whether E/\overline{k} is ordinary or supersingular.

We shall now consider some much more subtle moduli problems. As before, l is a prime not dividing d, and U ⊆ O_{D,f}^{×} is a compact open subgroup with properties (i)–(iii) above. We have seen that X^{D}(U) is a scheme over ℤ[1/M_{U}d]. We shall now “build” on this scheme by adding some structures at l, analogous to structures already considered on elliptic curves.

Definition 3.1. A Γ_{0}(l)-structure on a false elliptic curve A/S is an isogeny of false elliptic curves A/S → B/S of false degree l.

Remark. We have imposed no “cyclicity” condition in this definition, for simplicity. We can do this because l is prime. Compare the definitions of a Γ_{0}(l) structure on an elliptic curve in [DR] and in [KM], and also [KM] Corollary 6.8.7.

It is sometimes helpful to have an alternative, equivalent definition of a Γ_{0}(l)-structure, which we shall now describe. If A/S is a false elliptic curve, then A[l] is a finite Sgroup scheme with an action of O_{D} ⊗ 𝔽_{l}, which via the map κ of §1 can be identified

with M_{2}(𝔽_{l}). If e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(𝔽_{l}), and we write G_{1} = ker(e : A[l] → A[l]) and

G_{2} = ker(1 − e : A[l] → A[l]), then A[l] ≅ G_{1} × G_{2}.

If ψ : A → B is an isogeny of false elliptic curves over S, with false degree l, then ker ψ is a closed subgroup scheme of A[l], which is finite and locally free of rank l^{2}, and is fixed by the action of M_{2}(𝔽_{l}). Hence ker ψ ≅ K_{1} × K_{2} ⊆ G_{1} × G_{2}, where K_{1} is the kernel of e : ker ψ → ker ψ and K_{2} is the kernel of 1 − e on ker ψ. Note that K_{1}/S is finite and locally free of rank l, and isomorphic to K_{2}.

Conversely, if K_{1} is an arbitrary finite flat subgroup scheme of G_{1} which is locally free of rank l, then setting K_{2} to be the subgroup scheme of G_{2} which becomes identified

with K_{1} under the isomorphism of G_{1} with G_{2} induced by ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ∈ M_{2}(𝔽_{l}), we see that

K_{1} × K_{2} is an M_{2}(𝔽_{l})-invariant subgroup of A[l] of rank l^{2}, and ψ : A → A/(K_{1} × K_{2}) is an isogeny of abelian surfaces. Moreover, A/(K_{1} × K_{2}) can be given the structure of a false elliptic curve in such a way that ψ becomes an isogeny of false elliptic curves of false degree l.

We deduce that there are bijections between the Γ_{0}(l)-structures on A, the closed O_{D}-invariant subgroups of A[l] which are locally free of rank l^{2}, and the closed subgroups K_{1} of ker(e : A[l] → A[l]) which are locally free of rank l.

We now turn to some other structures, analogous to structures already defined on

elliptic curves. Recall that e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(𝔽_{l}) and if A/S is a false elliptic curve then

e acts on A[l]. Recall also from [KM] that if K is a finite flat group scheme over S which is locally free of rank n, then a point P ∈ K(S) is called a generator of K if the n points P, 2P, 3P, …, nP form a “full set of sections” for K ([KM] 1.8).

Let A/S be a false elliptic curve, where S is a ℤ[1/d]-scheme as usual. As usual, l is a prime not dividing d.

Definition 3.2. A Γ_{1}(l)-structure on A/S is

• π : A → B an isogeny of false elliptic curves, of false degree l,

• P ∈ K_{1}(S) a generator of K_{1} = ker(e : ker π → ker π).

Definition 3.3. A balanced Γ_{1}(l)-structure on (A/S, i) is the following collection of data:

• π : A → B an isogeny of false elliptic curves, of false degree l,

• π^{t} : B → A the dual isogeny.

• P ∈ K_{1}(S) a generator of K_{1} = ker(e : ker π → ker π),

• P′ ∈ K′_{1}(S) a generator of K′_{1} = ker(e^{∗} : ker π^{t} → ker π^{t}).

Now assume that S is a ℤ[1/d][ζ_{l}]-scheme, where ζ_{l} is a fixed primitive lth root of unity.

Definition 3.4. A canonical balanced Γ_{1}(l)-structure on (A/S, i) is the following collection of data:

• π : A → B an isogeny of false elliptic curves, of false degree l and kernel K,

• π^{t} : B → A the dual isogeny, with kernel K′,

• P ∈ K_{1}(S) a generator of K_{1} = ker(e : K → K),

• P′ ∈ K′_{1}(S) a generator of K′_{1} = ker(e^{∗} : K′ → K′), such that ⟨P, P′⟩_{π} = ζ_{l}. Here ⟨P, P′⟩_{π} is the canonical pairing between ker π and ker π^{t}.

Key remark. If S is a ℤ[1/d]-scheme, and A/S and B/S are false elliptic curves, such that A[l] ≅ B[l] as O_{D}-group schemes, then there is an induced isomorphism between the set of Γ_{0}(l) (resp. Γ_{1}(l), resp. balanced Γ_{1}(l), resp. canonical balanced Γ_{1}(l) if S/ℤ[1/d][ζ_{l}]) structures on A and on B. Moreover, if E/S is a (genuine) elliptic curve, and A[l] ≅ E[l]^{2} as O_{D}-group schemes, where here O_{D} acts via the natural action of O_{D} ⊗ 𝔽_{l} ≅ M_{2}(𝔽_{l}) on E[l]^{2}, then we have an induced isomorphism between the set of Γ_{0}(l) (resp. Γ_{1}(l), resp. etc) structures on A, as defined above, and the set of Γ_{0}(l) (resp. Γ_{1}(l), resp. etc) structures on E/S as defined in [KM]. The only structure for which this is not clear given the results of [KM] Chapter 1 is the Γ_{0}(l) structure, because of the cyclicity condition in [KM] for Γ_{0}(l) structures. But in the theory of elliptic curves, all l-isogenies are cyclic by [KM] Corollary 6.8.7.

§4. Analysis of the new structures.

We now wish to prove many results about these structures at l. All the results here are analogues of results proved in [DR]. Again throughout this section, l is a prime not dividing d.

Let S_{0} be a base scheme, and let F be a contravariant functor from false elliptic curves over S_{0}-schemes to the category of sets. For example, S_{0} = Spec(ℤ[1/d]) and F is the functor defined on objects by F(A/S) = {isomorphism classes of Γ_{0}(l)-structures on A/S}, or S_{0} = Spec(ℤ[1/d][ζ_{l}]) and F is the functor defined by F : A/S ↦ {isomorphism classes of canonical balanced Γ_{1}(l)-structures on A/S}. We say that F is relatively representable if, given any fixed S/S_{0} and any fixed false elliptic curve A/S, the functor from S-schemes to sets sending T/S to F(A_{T}) is representable. We now prove some relative representability results.

Proposition 4.1. If S_{0} = Spec(ℤ[1/d]), and T ∈ {Γ_{0}(l), Γ_{1}(l), balanced Γ_{1}(l) }, then the functor F defined by F(A/S) = {isomorphism classes of T -structures on A/S} is relatively representable. If S_{0} = Spec(ℤ[1/d][ζ_{l}]), then the functor F(A/S) = {isomorphism classes of canonical balanced Γ_{1}(l)-structures on A/S} is relatively representable.

Proof. Let A/S/ℤ[1/d] be a false elliptic curve. The result for F = Γ_{0}(l) is a consequence of the representability of a certain Grassmannian. For the map A[l] → S is finite and locally free, and hence defines a coherent sheaf A of bi-algebras on S, namely the pushforward of O_{A[l]}. A Γ_{0}(l)-structure can be thought of in this context as an O_{D}-invariant quotient B of A which is locally free of rank l^{2}, and such that the kernel of A → B is a bi-ideal of A. Hence F is represented by a closed subscheme of the Grassmannian of all rank l^{2} quotients of A.

In more concrete terms, the above argument shows the existence of a scheme S′/S together with a Γ_{0}(l)-structure ϖ : A_{S′} → B on A_{S′} which is universal for F, in the sense that every Γ_{0}(l)-structure on every extension A_{T} of A arises as a pullback of ϖ. Hence it is easy to see that the functor from S-schemes to sets sending T/S to the set of pairs consisting of a Γ_{0}(l)-structure π, and a homomorphism α from ℤ/lℤ to (ker(e : ker π → ker π))(T), is representable by ker(e : ker ϖ → ker ϖ). The relative representability of the Γ_{1}(l)-structure follows from this, because there is a closed subscheme of this scheme which is universal for the statement “α is a generator” by [KM] 1.9.1.

The relative representability of the balanced Γ_{1}(l) functor can be deduced in a similar way.

Finally, if S is a scheme over ℤ[1/d][ζ_{l}], A/S is a false elliptic curve, and S′′ represents the functor of balanced Γ_{1}(l) structures on A/S, then there is a closed subscheme of S′′ which is universal for the statement “the structure is canonical”.

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Corollary 4.2. If U is compact and open satisfying (i)–(iii) as usual, and l does not divide M_{U}d, then the functor from ℤ[1/M_{U}d]-schemes to sets sending S to the set of (isomorphism classes of) false elliptic curves over S equipped with a level U structure and a Γ_{0}(l)-structure is representable by a scheme X^{D}(U, Γ_{0}(l)).

Proof. Apply the relative representability result for Γ_{0}(l) to the universal false elliptic curve over X^{D}(U).

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Corollary 4.3. If U is as above, and l is a prime not dividing M_{U}d, then the functor from ℤ[1/M_{U}d][ζ_{l}]-schemes to sets sending S to the set of isomorphism classes of false elliptic curves over S, equipped with a level U structure and a canonical balanced Γ_{1}(l)- structure, is representable by a scheme X^{D}(U, Bal.can Γ_{1}(l))/ℤ[1/M_{U}d][ζ_{l}].

Proof. Apply the relative representability result for canonical balanced Γ_{1}(l) to the universal false elliptic curve over the pullback X^{D}(U)_{ℤ[1/M_{U} d][ζ_{l}]} of X^{D}(U).

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We now show that these new moduli problems are in fact extensions of naïve moduli problems discussed earlier.

Lemma 4.4.

(i) If S/ℤ[1/dl] is a scheme, and A/S is a false elliptic curve, then there is a bijection between the Γ_{0}(l)-structures on A and the naïve V_{0}(l)-structures on A. Similarly, there is a bijection between the Γ_{1}(l)-structures on A and the naïve V_{1}(l)-structures on A. (ii) If S/ℤ[1/dl][ζ_{l}] is a scheme, and A/S is a false elliptic curve, then there is a bijection between the canonical balanced Γ_{1}(l)-structures on A and the naïve V_{1}(l)-structures on A.

Proof.

(i) Firstly, we note that the functor on S-schemes sending T/S to the set of Γ_{0}(l)-structures on A_{T} is an étale sheaf on S, and the functor on S-schemes sending T/S to the set of naïve V_{0}(l)-structures on A_{T} is also an étale sheaf. Hence it suffices to establish the required bijection after replacing S by an étale surjective cover of S. So without loss of generality we may assume that A[l] ≅ ((ℤ/lℤ)^{4})_{S}. If α is a V_{0}(l) structure on A/S then we can choose

a full level l structure β : (O_{D} ⊗ ℤ/lℤ)_{S} → A[l] which lifts α. The subgroup β(( \begin{matrix}0 & 0 \\ 0 & ∗ \end{matrix} ))

of A[l] is finite and locally free of rank l, and killed by e, and as we showed above, this can be thought of as a Γ_{0}(l) structure on A/S. It is easily checked that this sets up the required bijection between V_{0}(l)-structures and Γ_{0}(l)-structures.

A similar argument gives us a bijection between naïve V_{1}(l)-structures and Γ_{1}(l)- structures on A/S. Explicitly, the point of exact order l corresponding to the naïve V_{1}(l)-

structure β is the point β(( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )).

(ii) By (i), it suffices to show that every Γ_{1}(l)-structure on A/S can be uniquely extended to a canonical balanced Γ_{1}(l)-structure. We see this as follows. Let π : A → B and P ∈ ker(e : ker π → ker π) be a Γ_{1}(l)-structure on A. Then the fact that the pairing ⟨, ⟩_{π} : ker π×ker π^{t} → μ_{l} is perfect and that P ≠ 0 implies that there exists P_{1} ∈ ker(π^{t})(S) such that ⟨P, P_{1}⟩_{π} = ζ_{l} ∈ μ_{l}(S). Let P′ = (1 − e^{∗})P_{1}. Then ⟨P, P′⟩ = ⟨P, (1 − e^{∗})P_{1}⟩ = ⟨(1 − e)P, P_{1}⟩ = ⟨P, P_{1}⟩ = ζ_{l}, and we deduce that Γ_{1}(l)-structures can be extended to canonical balanced Γ_{1}(l)-structures.

Moveover, because l is invertible on S, we see that this extension is unique. For if Q is any other S-valued point of ker(e^{∗} : ker π^{t} → ker π^{t}) then Zariski locally we have Q = nP′ for some integer n, and hence ⟨P, Q⟩ = (ζ_{l})^{n}. But ⟨P, Q⟩ = ζ_{l}, and we deduce that n is congruent to 1 mod l, and hence that Q = P′.

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We now prove an analogue of the Serre-Tate theorem, for false elliptic curves. In fact this theorem can be readily deduced from the Serre-Tate theorem for abelian varieties.

Theorem 4.5. Let l be a prime not dividing d. Let R be a ring in which l is nilpotent, let I ⊆ R be a nilpotent ideal, and set R_{0} = R/I. Let A_{0} be a false elliptic curve over Spec(R_{0}). Then there is a functorial bijection between the following two sets:

(a) Isomorphism classes of false elliptic curves A/R equipped with an isomorphism A × Spec(R_{0}) → A_{0}.

(b) Isomorphism classes of l-divisible groups G/R equipped with an action of O_{D} (or equivalently an action of M_{2}(ℤ_{l})) and an O_{D}-isomorphism G × Spec(R_{0}) → A_{0}[l^{∞}]. The bijection is given by the map A ↦ A[l^{∞}].

Proof. It suffices to write down an inverse to this map. Given an l-divisible group as in (b), the Serre-Tate theorem (see for example Theorem 1.2.1 of [Ka]) tells us that there is some abelian surface A/R such that A×Spec(R_{0}) is isomorphic to A_{0}. It remains to put an action of O_{D} on A, lifting the action on A_{0}. Again we can do this using the Serre-Tate theorem, which implies that if j ∈ O_{D} then the endomorphisms induced by j on G and on A_{0} can be glued uniquely to give an endomorphism of A.

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Corollary 4.6. Let A/\overline{𝔽}_{l} be a false elliptic curve, and let E/\overline{𝔽}_{l} be an elliptic curve, such that E is ordinary if A is ordinary, and supersingular if A is supersingular. Let W be the Witt vectors of \overline{𝔽}_{l}, let 𝔸/W [[T]] be the universal formal deformation of A (see [B] §5 Corollaire) and let 𝔼/W [[T]] be the universal formal deformation of E.

(i) 𝔸[l^{∞}] is the universal formal deformation of A[l^{∞}] (to l-divisible groups with an O_{D}-action over Artin local W -algebras).

(ii) 𝔸[l^{∞}] ≅ 𝔼[l^{∞}]^{2} where O_{D} acts on 𝔼[l^{∞}]^{2} via the canonical action of O_{D} ⊗ ℤ_{l} ≅ M_{2}(ℤ_{l}).

Proof. (i) is immediate from Theorem 4.5. For (ii) it suffices to note that deforming A[l^{∞}] with its O_{D}-action is equivalent to deforming A[l^{∞}] with its induced M_{2}(ℤ_{l})-action,

which is equivalent to deforming ker(ẽ : A[l^{∞}] → A[l^{∞}]) where ẽ = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(ℤ_{l}).

Finally, by our choice of E, we have E[l^{∞}]^{2} ≅ A[l^{∞}], E[l^{∞}] ≅ ker(ẽ : A[l^{∞}] → A[l^{∞}]), and hence by the Serre-Tate theorem again we deduce that the deformation theories of A, A[l^{∞}] (with their O_{D}-actions), E and E[l^{∞}] are all equivalent in this setting.

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We will use this latter corollary to deduce certain results about Shimura curves from the corresponding results for modular curves. The next theorem is an example of this.

As in [DR] we use the suffix ^{h} to denote the ordinary (at l) locus of a moduli problem. So for example X^{D}(U, Γ_{0}(l))^{h} is the scheme obtained by removing the supersingular points of X^{D}(U, Γ_{0}(l)) in the fibre at l.

The following theorem summarises the main results which we shall need about the scheme X^{D}(U, Γ_{0}(l)).

Theorem 4.7.

(i) The scheme X^{D}(U, Γ_{0}(l)) is smooth over ℤ[1/M_{U}d] away from the supersingular points in characteristic l. (ii) X^{D}(U, Γ_{0}(l))/ℤ[1/M_{U}d] is proper. (iii) The map c : X^{D}(U, Γ_{0}(l)) → X^{D}(U) forgetting the Γ_{0}(l) structure is finite. (iv) c is flat, and X^{D}(U, Γ_{0}(l)) is a regular scheme.

(v) The fibre X^{D}(U, Γ_{0}(l))_{𝔽_{l}} of X^{D}(U, Γ_{0}(l)) is composed of two irreducible components, both isomorphic to X^{D}(U)_{𝔽_{l}}, which cross transversally at the supersingular points.

Remark. The proofs of the analogous results in [DR] V.1 go through in many cases. We shall also use the techniques of [KM] Chapter 5.

Proof.

(i) Let A(U) denote the functor from ℤ[1/M_{U}d]-schemes to sets, sending S to the set of false elliptic curves over S equipped with a naïve level U structure and an O_{D}-invariant subgroup scheme which is locally (which for the duration of this paragraph means “locally for the étale topology”) O_{D}-isomorphic to (μ_{l})^{2} with its canonical action of O_{D} (via the natural action of M_{2}(𝔽_{l}) and the isomorphism O_{D} ⊗ 𝔽_{l} ≅ M_{2}(𝔽_{l}) fixed in section §1). Define B(U) to be the functor from ℤ[1/M_{U}d]-schemes to sets, sending S to the set of false elliptic curves over S equipped with a level U structure and a closed O_{D}-invariant subgroup scheme locally isomorphic to (ℤ/lℤ)^{2}. If (A/S, i) is a false elliptic curve with a naïve level U structure α and a closed O_{D}-invariant subgroup scheme G of A[l] locally isomorphic to (μ_{l})^{2}, then A/G can also be given the structure of a false elliptic curve over S, and the O_{D}- invariant subgroup scheme A[l]/G is locally isomorphic to (ℤ/lℤ)^{2}. Moreover, the naïve level U structure on A can be pushed forward to A/G. This induces a natural isomorphism of functors from A(U) to B(U). Moreover, by the theory of the Hilbert scheme, B(U) is representable, and by SGA 3, IX 3.6 bis, A(U) is formally étale over X^{D}(U) and hence smooth over ℤ[1/M_{U}d]. So both A(U) and B(U) are smooth over ℤ[1/M_{U}d]. Moreover, X^{D}(U, Γ_{0}(l))^{h} is the union of A(U) and B(U), and furthermore the intersection of A(U) and B(U) is the open subscheme of X^{D}(U, Γ_{0}(l))^{h} obtained by removing the fibre at l, which is nothing other than the scheme X^{D}(U ∩V_{0}(l)) by Lemma 4.4. Hence X^{D}(U, Γ_{0}(l)) is smooth away from the supersingular points. (ii) This follows from the valuative criterion of properness and the fact that false elliptic curves have potentially good reduction (see section 6 of [B]). One need only check that if A/O is an abelian scheme over a discrete valuation ring, and π is a Γ_{0}(l) structure on the generic fibre of A, that π extends uniquely to a Γ_{0}(l) structure on A. This follows from the fact that A is a Néron model of its generic fibre. (iii) The map c is proper, and so it suffices to check that its geometric fibres are finite. In particular, it suffices to show that if A/k is a false elliptic curve over an algebraically closed field, then there are only finitely many Γ_{0}(l)-structures on A/k. We have A[l] ≅ G × G, for G of rank l^{2}, and a Γ_{0}(l) structure on A/k can be thought of as a rank l subgroup of G. There are three cases, which can be checked separately:

(a) The characteristic of k is not equal to l. In this case, G is isomorphic to (ℤ/lℤ)^{2} and there are l + 1 subgroups of G having rank l.

(b) char(k) = l and A/k ordinary. Then G is isomorphic to ℤ/lℤ × μ_{l} and there are only 2 subgroups of G of rank l, namely ℤ/lℤ and μ_{l}.

(c) char(k) = l and A/k supersingular. Then G ≅ α_{l^{2}} because G is the l-torsion in the unique one-parameter formal Lie group of height 2 over k. Hence there is only one subgroup of G of rank l, namely α_{l}, the kernel of Frobenius. Thus c is proper with finite fibres and is hence finite. (iv) Here, rather than modifying proofs in [DR] and [KM] into this setting, we instead actually use results of [KM] to deduce results in our case. In particular, [KM] have to work quite hard to get their Theorem 6.6.1, deducing it from analogous results for Γ_{1}(l) structures. However, we do not have to redo the results of [KM] chapters 5 and 6 in our setting, because we can use [KM] Theorem 6.6.1 itself. Firstly here is a brief summary of what we shall do.

The conditions that we have to check (regularity and flatness) are conditions on the local rings of X^{D}(U) and X^{D}(U, Γ_{0}(l)). Firstly we reduce the problem to having to check these conditions only on the local rings of closed points with residue field of characteristic l. Then we show that, after strict henselisation and completion (which does not affect the properties we need to check), these latter local rings become isomorphic to the local rings of points on modular curves, where flatness and regularity have already been proved in [DR] or [KM]. Hence we deduce flatness and regularity in our case. Now here are the details.

Consider the subset Z of X^{D}(U) consisting of the points y ∈ X^{D}(U) such that, for all points x of X^{D}(U, Γ_{0}(l)) lying above y, the local ring O_{x} is regular, and flat over O_{y}. Then Z is open, as c is a closed map. We must show that Z = X^{D}(U). Now because X^{D}(U) is of finite type over ℤ[1/M_{U}d], we need only prove that Z contains every closed point of X^{D}(U). For points of residue characteristic not equal to l this is a consequence of the results of §2, where a Γ_{0}(l) structure can be interpreted as a naïve V_{0}(l) structure. Hence we are reduced to considering points of residue characteristic l.

Hence we must show that if y ∈ X^{D}(U) is a closed point of residue characteristic l, and x ∈ X^{D}(U, Γ_{0}(l)) lies above it, then the local ring O_{x} is regular, and flat over O_{y}. We shall do this by deducing it from the corresponding result for modular curves, using the ideas on p133 of [KM]. Firstly we reduce the problem to one about complete strictly Henselian local rings. Let k = \overline{𝔽}_{l}, and let W be the Witt vectors of k. For a local ring A, let A^{h.s.} denote its strict henselisation, and Â its completion.

Let y_{0} be any closed point of X^{D}(U) of residue characteristic l, and let y be any k-valued point of X^{D}(U) ×_{ℤ} W lying over y_{0}. Then the local rings of y and y_{0} are related by { \atop O_{X^{D}(U),y_{0}}^{h.s.}} ≅ { \atop O_{X^{D}(U)×W,y}}, and it follows from this fact that we are done if we can show that for any k-valued point x of X^{D}(U, Γ_{0}(l)) × W lying over y, the complete local ring

{ \atop O_{X^{D}(U,Γ_{0}(l))×W,x}} is regular, and flat over { \atop O_{X^{D}(U)×W,y}}.

Because c is finite from (iii), this can be reinterpreted as saying that the scheme

(X^{D}(U, Γ_{0}(l)) × W) ×_{(X^{D}(U)×W)} Spec({ \atop O_{X^{D}(U)×W,y}})  (∗)

is regular, and flat over { \atop O_{X^{D}(U)×W,y}}. This scheme is nothing other than the scheme relatively representing Γ_{0}(l)-structures on the false elliptic curve 𝔸/{ \atop O_{X^{D}(U)×W,y}}, the universal formal deformation of the false elliptic curve A/k corresponding to y. Now let E/k be an elliptic curve which is ordinary if A is ordinary, and supersingular if A is supersingular. By Corollary 4.6, 𝔸[l] ≅ 𝔼[l]^{2} where 𝔼 is the universal formal deformation of E/k to Artin local W -algebras with residue field k. We deduce that to give a Γ_{0}(l)-structure on 𝔸 is to give one on 𝔼, and hence the scheme (∗) above is isomorphic to the corresponding scheme denoted (∗) on p133 of [KM], for P = [Γ_{0}(l)] in the notation of [KM]. Now by [KM] Theorem 6.6.1, this scheme is indeed a regular scheme, flat over { \atop O_{X^{D}(U)×W,y}} (which is isomorphic to W [[T]] by [B] §5 and Proposition 2.5) and hence Z does indeed contain every closed point with residue characteristic l and we are done.

(v) If S is a scheme over 𝔽_{l}, then the absolute Frobenius map F_{abs} : S → S gives us by pullback a diagram

A^{(l)} ⟶ A
↓ ↓
S ⟶^{F_{abs}} S

and hence in the usual way an S-morphism F : A → A^{(l)}. The endomorphisms i(x) of A, for x ∈ D, induce endomorphisms of A^{(l)}, and these endomorphisms define the structure of a false elliptic curve on A^{(l)}, with respect to which F becomes an isogeny of false elliptic curves, of false degree l. Write V for the dual of F.

There are maps Φ_{1} and Φ_{2} : X^{D}(U)_{𝔽_{l}} → X^{D}(U, Γ_{0}(l))_{𝔽_{l}} defined (functorially) by

Φ_{1} : (A, ϕ) ↦ (A, ϕ, F)

and

Φ_{2} : (A, ϕ) ↦ (A^{(l)}, F_{∗}ϕ, V),

where F_{∗}ϕ denotes the push-forward of ϕ defined in §1. Hence we have a map

Φ_{1} ⨿ Φ_{2} : X^{D}(U)_{𝔽_{l}} ⨿ X^{D}(U)_{𝔽_{l}} → X^{D}(U, Γ_{0}(l))_{𝔽_{l}}.

If c : X^{D}(U, Γ_{0}(l)) → X^{D}(U) is the forgetful map, then cΦ_{1} is the identity on X^{D}(U)_{𝔽_{l}}, and cΦ_{2} is the map (A, ϕ) ↦ (A^{(l)}, F_{∗}ϕ).

Define w : X^{D}(U, Γ_{0}(l)) → X^{D}(U, Γ_{0}(l)) thus: If A/S is a false elliptic curve over a ℤ[1/M_{U}d]-scheme, with a level U structure ϕ and an isogeny ψ : A → B of false degree l, then w(A, ϕ, ψ) = (B, ψ_{∗}ϕ, ψ^{t}). Note that Φ_{2} = wΦ_{1}.

The two maps Φ_{1} and Φ_{2} are closed immersions, because they have left inverses. Moreover, the only rank l subgroups of (μ_{l} × ℤ/lℤ)/\overline{𝔽}_{l} are μ_{l} and ℤ/lℤ, and hence

Φ_{1} ⨿ Φ_{2} : X^{D}(U)_{𝔽_{l}}^{h} ⨿ X^{D}(U)_{𝔽_{l}}^{h} → X^{D}(U, Γ_{0}(l))_{𝔽_{l}}^{h}

is an isomorphism. Hence the images of Φ_{1} and Φ_{2} are the two irreducible (by Proposition 2.4) components of X^{D}(U, Γ_{0}(l))_{𝔽_{l}}. Finally we must show that the intersection of these two components at the supersingular points is transversal. One way of seeing this is as follows: we have a map c : X^{D}(U, Γ_{0}(l)) → X^{D}(U) forgetting the Γ_{0}(l) structure. The map cw is hence also a map from X^{D}(U, Γ_{0}(l)) to X^{D}(U). Now one sees that the tangent vectors at a supersingular point in characteristic l are distinct, because the one corresponding to Φ_{1} is killed by d(cw) and not by d(c), whereas the one corresponding to Φ_{2} is killed by is killed by d(c) and not by d(cw).

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Another method of analysing the supersingular points is by reducing to the classical case, as follows. The spectrum of the completed local ring at any supersingular point x of X^{D}(U, Γ_{0}(l))_{\overline{𝔽}_{l}} is the scheme relatively representing Γ_{0}(l)-structures on the universal deformation of A/\overline{𝔽}_{l} to Artin local \overline{𝔽}_{l}-algebras, where A/\overline{𝔽}_{l} is the false elliptic curve corresponding to x. But this scheme is isomorphic to the scheme relatively representing Γ_{0}(l)-structures on the universal deformation of a supersingular elliptic curve over \overline{𝔽}_{l} to Artin local \overline{𝔽}_{l}-algebras, which is the spectrum of the completed local ring of the classical modular curve classifying elliptic curves with a Γ_{0}(l) structure and some sufficiently strong naïve level structure prime to l. Hence one can again read off certain results on the structure of these complete local rings from classical results.

Before we embark upon an analysis of the structure of X^{D}(U, Bal.can Γ_{1}(l)) we have to define one final structure, which only makes sense for false elliptic curves over 𝔽_{l}-schemes.

Recall that if S/𝔽_{l} is a scheme and A/S is a false elliptic curve, then the Frobenius morphism F : A → A^{(l)} is an isogeny of false elliptic curves, of false degree l, and V is defined to be the isogeny dual to F. We have that ker(V) splits up as H_{1} ×_{S} H_{2}, with H_{1} = ker(e : ker V → ker V) and H_{2} = ker(1 − e : ker V → ker V) as usual.

Definition 4.8. An Igusa structure on (A/S/𝔽_{l}, i) is a point P ∈ H_{1}(S) which is a generator of H_{1} = ker(e : ker V → ker V).

Proposition 4.9. Igusa structures are relatively representable, i.e. if A/S/𝔽_{l} is a false elliptic curve, then the functor from S-schemes to sets sending T/S to the set of isomorphism classes of Igusa structures on A_{T} is representable.

Proof. Indeed, the functor is represented by a closed subscheme of ker(e : ker V → ker V).

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As a corollary we see that if l ∤ M_{U} then there is a scheme X^{D}(U, Ig(l)) classifying isomorphism classes of false elliptic curves over 𝔽_{l}-schemes equipped with a naïve level U structure and an Igusa structure at l. Analogously to the classical case, the morphism X^{D}(U, Ig(l)) → X^{D}(U)_{𝔽_{l}} is étale over the ordinary locus and totally ramified over the supersingular points. One can see this as follows. If x is an ordinary point of X^{D}(U, Ig(l))_{\overline{𝔽}_{l}} with residue field \overline{𝔽}_{l} and y is its image in X^{D}(U)_{\overline{𝔽}_{l}}, then the completed local rings of x and y are isomorphic, by reduction to the classical case. Finally, if A/\overline{𝔽}_{l} is a supersingular false elliptic curve, then there is a unique Igusa structure on A, namely the origin. As a corollary, one deduces that X^{D}(U, Ig(l)) is connected (because supersingular false elliptic curves exist). Moreover, X^{D}(U, Ig(l)) is regular (again we can see this immediately by reducing to the classical case, because a Noetherian local ring is regular if and only if its completion is regular), and hence irreducible. Lastly, the map X^{D}(U, Ig(l)) → X^{D}(U) is flat, again because of the analogous result in the classical case.

The second main theorem of this section is the following analogue of [DR] Theorem V.2.12. Recall that we have shown in Corollary 4.3 that if U is a compact open subgroup of O_{D,f}^{×} satisfying the usual conditions (i)–(iii), then the functor sending an ℤ[1/M_{U}d][ζ_{l}]-scheme S to the set of false elliptic curves (A/S, i) equipped with a naïve level U structure and a canonical balanced Γ_{1}(l)-structure is representable by a scheme X^{D}(U, Bal.can Γ_{1}(l))/ℤ[1/M_{U}d][ζ_{l}].

Theorem 4.10.

(i) X^{D}(U, Bal.can Γ_{1}(l))^{h} → Spec(ℤ[1/M_{U}d][ζ_{l}]) is smooth of relative dimension 1. (ii) X^{D}(U, Bal.can Γ_{1}(l)) → Spec(ℤ[1/M_{U}d][ζ_{l}]) is proper. (iii) The natural map X^{D}(U, Bal.can Γ_{1}(l)) → X^{D}(U, Γ_{0}(l))_{ℤ[1/M_{U} d][ζ_{l}]} is finite, and hence X^{D}(U, Bal.can Γ_{1}(l)) → X^{D}(U)_{ℤ[1/M_{U} d][ζ_{l}]} is also finite. (iv) The morphism X^{D}(U, Bal.can Γ_{1}(l)) → X^{D}(U, Γ_{0}(l))_{ℤ[1/M_{U} d][ζ_{l}]} is flat, and the scheme X^{D}(U, Bal.can Γ_{1}(l)) is regular.

(v) The fibre X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}} is composed of two irreducible components, both isomorphic to X^{D}(U, Ig(l)), and which cross transversally at the supersingular points.

Proof. Many of the ideas here are similar to those in the proof of Theorem 4.7.

(i) It suffices to prove that X^{D}(U, Bal.can Γ_{1}(l))^{h} → X^{D}(U)^{h} is étale. Recall A(U) and B(U) defined in the course of the proof of Theorem 4.7. Let U(U) (resp. V(U)) be the functor on ℤ[1/M_{U}d][ζ_{l}]-schemes sending a scheme S to the set of isomorphism classes of false elliptic curves A/S equipped with a level U structure and a canonical balanced Γ_{1}(l)- structure (π, π^{t}, P, P′) such that ker π is locally (étale) O_{D}-isomorphic to μ_{l}^{2} (resp. (ℤ/lℤ)^{2}) with its natural O_{D}-action (via M_{2}(𝔽_{l})). Then the natural map U(U) → A(U)_{ℤ[1/M_{U} d][ζ_{l}]} (resp. V(U) → B(U)_{ℤ[1/M_{U} d][ζ_{l}]}) is finite and étale, and X^{D}(U, Bal.can Γ_{1}(l)) is the union of U(U) and V(U), glued over U(U)[1/l] ≅ V(U)[1/l] where ℤ/lℤ and μ_{l} become canonically isomorphic via the map 1 ↦ ζ_{l}. (ii) Again this follows from the valuative criterion of properness, once one checks that canonical balanced Γ_{1}(l) structures extend uniquely from the generic fibre to the whole of a discrete valuation ring, after base extension if necessary. More precisely, the theorem of potentially good reduction in section 6 of [B] reduces us to showing the following: if K is a field and a ℤ[1/d][ζ_{l}]-algebra, and K is equipped with a discrete valuation, O_{K} is the integers of K, and A/O_{K} is a false elliptic curve, then a canonical balanced Γ_{1}(l)- structure on A_{K} extends uniquely to a canonical balanced Γ_{1}(l) structure on A. That the structure extends uniquely to a balanced Γ_{1}(l)-structure is immediate from the Néron property of A/O_{K}. Now one observes that the pairings defined in [O] are compatible with base extension, and hence that the extended structure is still canonical.

(iii) It is an easy check that X^{D}(U, Bal.can Γ_{1}(l)) → X^{D}(U, Γ_{0}(l))_{ℤ[1/M_{U} d][ζ_{l}]} is proper with finite fibres and is hence finite. (iv) As in Theorem 4.7 we can deduce regularity and flatness from the corresponding results in the modular curve case.

(v) The generic fibre of X^{D}(U, Bal.can Γ_{1}(l)) is isomorphic to X^{D}(U ∩ V_{1}(l))_{ℚ} × ℚ(ζ_{l}), where X^{D}(U ∩ V_{1}(l))_{ℚ} is the canonical model defined by Shimura. We can identify the special fibre at l of X^{D}(U, Bal.can Γ_{1}(l)) with the union of two copies of X^{D}(U, Ig(l))_{𝔽_{l}} by giving two morphisms from X^{D}(U, Ig(l))_{𝔽_{l}} to X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}} in the following way. The morphisms are induced by the two morphisms of moduli problems

j_{1} : (A/S, ϕ, P ∈ A^{(l)}(S)) ↦ (A/S, ϕ, F, V, 0, τ(P))

and

j_{2} : (A/S, ϕ, P ∈ A^{(l)}(S)) ↦ (A^{(l)}/S, F_{∗}ϕ, V, F, P, 0),

where τ is an appropriate matrix, for example τ = (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) t̄)^{−1} ∈ GL_{2}(𝔽_{l}), where t̄ is

the image of the t of §1 in M_{2}(𝔽_{l}). Note that

j_{1} ⨿ j_{2} : X^{D}(U, Ig(l)) ⨿ X^{D}(U, Ig(l)) → X^{D}(U, Bal.can Γ_{1}(l))_{𝔽_{l}}

is surjective on \overline{𝔽}_{l}-valued points, by checking the two cases. If A/\overline{𝔽}_{l} is an ordinary false elliptic curve, then the finite group scheme A[l] is isomorphic to (ℤ/lℤ × μ_{l})^{2} and hence at least one of P and P′ must be the zero section. If A/\overline{𝔽}_{l} is supersingular, then F = V and P = P′ = 0 is the only canonical balanced Γ_{1}(l)-structure on A and we see that this structure is in the image of both j_{1} and j_{2}. Moreover, j_{1} and j_{2} are certainly monomorphisms by their definitions, and hence closed immersions because they are proper, being X^{D}(U)-maps between finite flat X^{D}(U)_{𝔽_{l}}-schemes (after choosing appropriate maps down to X^{D}(U)). We deduce that the special fibre at l of X^{D}(U, Bal.can Γ_{1}(l)) has two irreducible components, which intersect at the supersingular points. One can again check that the intersection at the supersingular points is transversal by using the analogous result for the modular curve case.

□

Proofs of analogous results for totally real fields other than ℚ have been given by Carayol in [C] and by Jarvis in [J].

If we set U = V_{1}(N) for some N ≥ 4 then X^{D}(U, Bal.can Γ_{1}(l))_{ℤ_{l}[ζ_{l}]} is an analogue of the well-known regular model of X_{1}(Nl) over ℤ_{l}[ζ_{l}]. We see from the above theorem and [DR] that these two curves have very similar properties in characteristic l. There is one aspect of the theory of these Shimura curves where the analogy with modular curves is less than perfect. There is no need for a theory of compactification in this setting, because the Shimura curves over ℂ arising from indefinite non-split quaternion algebras are already compact Riemann surfaces. Hence there are no cusps in this setting. Whilst saving us the details of compactification, it also means that there is no obvious notion of a q-expansion.

§5. The Hasse invariant.

As an example of how the analogy with modular curves continues, we shall construct a section a ∈ H^{0}(X^{D}(V_{1}(N), Ig(l))_{𝔽_{l}}, ω) which plays the role of the section a defined in §5 of [G], and in [KM].

Define X_{1}^{D}(N)_{ℤ_{l}} := X^{D}(V_{1}(N))_{ℤ_{l}} and I_{1}^{D}(N)_{𝔽_{l}} := X^{D}(V_{1}(N), Ig(l))_{𝔽_{l}}. Recall from §4 that there is a natural map π : I_{1}^{D}(N)_{𝔽_{l}} → X_{1}^{D}(N)_{𝔽_{l}} forgetting the Igusa structure, which is étale away from the supersingular points, and totally ramified of degree l − 1 at the supersingular points. In [DT] §4 an invertible sheaf ω = ω_{X_{1}^{D}(N)_{ℤ_{l}}} is defined on X_{1}^{D}(N)_{ℤ_{l}}, analogous to the sheaf ω_{X_{1}(N)_{ℤ_{l}}} on X_{1}(N)_{ℤ_{l}} whose sections give us modular forms. If pr : A_{univ} → X_{1}^{D}(N)_{ℤ_{l}} is the universal false elliptic curve over X_{1}^{D}(N)_{ℤ_{l}}, then

ω_{X_{1}^{D}(N)_{ℤ_{l}}} = ẽpr_{∗}Ω_{A_{univ}/X_{1}^{D}(N)_{ℤ_{l}}}^{1} , where ẽ = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ M_{2}(ℤ_{l}) ≅ O_{D}⊗ℤ_{l}. Define ω_{X_{1}^{D}(N)_{𝔽_{l}}}

to be the pullback of this sheaf to the special fibre, and define ω_{I_{1}^{D}(N)} to be π^{∗}ω_{X_{1}^{D}(N)_{𝔽_{l}}} . Let ss denote the divisor of supersingular points on I_{1}^{D}(N), and let s be the degree of this divisor, so s is the number of points in I_{1}^{D}(N)(\overline{𝔽}_{l}) (or equivalently in X_{1}^{D}(N)(\overline{𝔽}_{l})) corresponding to supersingular false elliptic curves over \overline{𝔽}_{l}.

Proposition 5.1. The invertible sheaf ω_{I_{1}^{D}(N)} on I_{1}^{D}(N) has degree s. We have
Ω_{I_{1}^{D}(N)}^{1} ≅ ω_{I_{1}^{D}(N)}^{⊗2}((l − 2)ss).

Proof. From Lemma 6 of [DT], we have 2s = (l−1) deg Ω_{X_{1}^{D}(N)/𝔽_{l}}, and from Lemma 7 of [DT] we deduce ω_{X_{1}^{D}(N)}^{⊗2} ≅ Ω_{X_{1}^{D}(N)/𝔽_{l}} (We remark that in [DT] l is assumed to be odd, and ẽ is chosen in M_{2}(ℤ_{l}) such that ẽ^{∗} = ẽ. This may not be possible if l = 2, but in any case it is sufficient for the proof of [DT] Lemma 7 that ẽ^{∗} be conjugate to ẽ, which is always true regardless of the value of l). Hence

\begin{aligned}2s & = (l − 1) deg Ω_{X_{1}^{D}(N)} \\ & = (l − 1)(2 deg ω_{X_{1}^{D}(N)}) \end{aligned}

and so deg ω_{I_{1}^{D}(N)} = (l − 1) deg ω_{X_{1}^{D}(N)} = s.

Now by the Riemann-Hurwitz formula, we have

Ω_{I_{1}^{D}(N)} ≅ π^{∗}Ω_{X_{1}^{D}(N)}((l − 2)ss)
≅ π^{∗}(ω_{X_{1}^{D}(N)}^{⊗2})((l − 2)ss)
≅ ω_{I_{1}^{D}(N)}^{⊗2}((l − 2)ss).
□

The following is the analogue of the section a defined in [G] §5.

Theorem 5.2. There is a non-zero global section a of ω_{I_{1}^{D}(N)} with simple zeros at precisely the supersingular points.

Proof. We define a analogously to [KM] Section 12.8. Firstly we construct, given a false elliptic curve (A/S, i) over S/𝔽_{l} with an Igusa structure P ∈ ker(V |_{A^{(l)}})(S), an element of H^{0}(A, Ω_{A/S}), thus. The point P gives rise, by duality, to a map

ϕ_{P} : ker(F |_{A}) → 𝔾_{m}.

The invariant differential dX/X on 𝔾_{m} gives rise by pullback to an invariant differential ϕ_{P}^{∗} (dX/X) on ker(F), which is the reduction of a unique invariant differential ν on A. This construction gives us, for any (A, i) over S/𝔽_{l} with an Igusa structure P, an element of H^{0}(A, Ω_{A/S}) = H^{0}(S, pr_{∗}Ω_{A/S}). Applying ẽ to this section gives us a section of H^{0}(S, ẽpr_{∗}Ω_{A/S}). If S = I_{1}^{D}(N) with its universal false elliptic curve (A, i, ϕ, P), then this construction defines the global section a which we seek. Note that if A/\overline{𝔽}_{l} is supersingular then P = 0 and hence the map ϕ_{P} is 0, so a vanishes at every supersingular point. On the other hand, if A/\overline{𝔽}_{l} is ordinary, then the construction gives us a non-zero element of H^{0}(S, ẽpr_{∗}Ω_{A/S}) and hence a is not the zero section of ω_{I_{1}^{D}(N)}.

To show that a has a simple zero at each supersingular point, we use a counting argument. We have shown that a has zeros at every supersingular point, and by Proposition 5.1 the degree of ω_{I_{1}^{D}(N)} is equal to s, the number of supersingular points. Hence a must have a simple zero at each supersingular point.

□

In fact, many of the geometrical constructions and results of [G] and [E] may be generalised to this setting. See [Bu] where more examples are given.

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